Chop the peeled and pitted peaches into 1/2 inch pieces. Place in a small saucepan with the honey. Bring the mixture to a boil, then reduce heat to medium low. Cover the pot and cook, stirring occasionally, just until the peaches are soft, about 10 minutes. Remove from heat and carefully blend using an immersion blender or stand blender.
In a larger saucepan, combine the milk and cream over medium heat.
In a medium bowl, beat the egg yolks with the sugar. While continuing to beat, slowly pour in a tablespoon of the heated milk and cream. Once incorporated, slowly stream in and whisk another 1/2 cup. Slowly pour the mixture back into the saucepan over medium heat while whisking. Cook while continuing to stir until just thickened enough to coat the back of a spoon, 170-180 degrees F. Remove from heat and transfer to a heat-safe bowl. Mix in the pureed peach and vanilla extract. Cover and refrigerate until well chilled, a few hours to overnight.
Transfer the chilled mixture to an ice cream machine and churn according to manufacturer's instructions. Once thickened and creamy, transfer to a freezer-safe container and freeze until firm, about 2 hours.